lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: 45e040d1a001023d7831dd72a1dc8acc2c579511
Parent: 841a1a48a14c12e33c2306032057b424e564b5f9
Author: Pablo <pablo-escobar@riseup.net>
Date: Sat, 17 Dec 2022 22:04:21 +0000

Clarified the PBW theorem

Added a remark on the fact that the basis we work with are all ordered

Diffstat

1 file changed, 4 insertions, 3 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	7	4	3

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -655,9 +655,10 @@ we find\dots
 
 \begin{theorem}[Poincaré-Birkoff-Witt]
   Let \(\mathfrak{g}\) be a Lie algebra over \(K\) and \(\{X_i\}_i \subset
-  \mathfrak{g}\) be a basis for \(\mathfrak{g}\). Then \(\{X_{i_1} \cdot
-  X_{i_2} \cdots X_{i_n} : n \ge 0, i_1 \le i_2 \le \cdots \le i_n\}\) is a
-  basis for \(\mathcal{U}(\mathfrak{g})\).
+  \mathfrak{g}\) be an orderer basis for \(\mathfrak{g}\) -- i.e. a basis
+  indexed by an ordered set. Then \(\{X_{i_1} \cdot X_{i_2} \cdots X_{i_n} : n
+  \ge 0, i_1 \le i_2 \le \cdots \le i_n\}\) is a basis for
+  \(\mathcal{U}(\mathfrak{g})\).
 \end{theorem}
 
 The Poincaré-Birkoff-Witt Theorem is hugely important and will come up again